PROPERTIES OF THE l'4'ljmerical FliNCTION Fs

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1 PROPERTIES OF THE l'4'ljmerical FliNCTION Fs by I. Blilceoiu, V. Seleacu, N. Virla Departamet of Mathematics, Uiversity of Craiova Craiova (1100), ROMANIA I this paper are studied some properties of the umerical fuctio Fs(x):N - {O,I} 4' N Fs(x) = L Sp(x), where 5 p (x) = S(p%) is the Smaradache O<pS.r p pme fuctio defied i [4]. Numerical example: Fs(5) = S(2 5 ) ~ 5(3 5 ) + S(5 5 ); Fs(6) = S(2 6 ) + 5(3 6 ) + 5(~). It is kow that: (p -1)r + 1 $ 5(pr) $ pr so(p -1)r < S(pr) $ pr. Tha (1) Where Jr( x) is the umber of prime umbers smaller or equal with x.. % 1 PROPOSITION 1: The sequece T( r) = 1 log Fs (x) + L --. has limit i=2 F:(l) Proof The iequality Fs(x) > r(p2+ 4- P:..r) - Jr(x» implies -logfs(x) < < -logr(a. + P P1I(%) - (r» < -logr((r)a. - ;r(r» = -logr -log 1Z'(x) -log(a. -1). Tha for x=i the iequality (1) become: < < =---- FS(i) i(pl +"'+P:(I) - Jr(i» i(pijr(i) - Jr(i» ijr(i)(pt -1).r 1 Tha T(x)<l-log(x)-logJr(x)-log(A-l)+L =2 m(i)(pt -1).r 1 A =2 =:- T(x) = l-logx-logjr(x)+ L =2 ljr(l) =:- lim T(x) $ 1 - lim logx-lim 10gJr(x) + lim i: -. _1_. =1-oo-oo+L=-oo. ;c~"" ;c... "" ;c... ~ ;c... oc 1=2 I Jr( I) PRoPOsmON 1. The equatio Fs(x) = Fs(x + 1) has o solutio for x en - to, I}. 6

2 Proof First we cosider that x-i is a prime umber with x > 2. I the particular case x = 2 we obtai Fs(2) = 5(22) = 4; Fs(3) = 5(2 3 ) + 5(3 3 ) = 4 T 9 = l3. So F 2 (2) < Fs(3). Next we shall write the iequalities: (2) U sig the reductio ad absurdum method we suppose that the equatio Fs (x) = Fs (x + 1) has solutio. From (2) results the iequalities From (3) results that: +7Z{x + 1) > 0. But Ptr(x+l) > 7Z{x + 1) so the diferece from above is egative for x> 0, ad we obtaied a cotradictio. So Fs ( x) = Fs ( x + 1) has o solutio for x + 1 a prime umber. Next, we demostrate that the equatio Fs ( x) = Fs (x + 1) has o solutio for x ad x + 1 both composite umbers. Let p be a prime umber satisfig coditios P > x ad P ~ x - I. Such P exists 2 accordig to Bertrad's postulate for every x E ~ - {O, I}. Tha i the factorial of the umber p( x-i), the umber p appears at least x times. So, we have S(pX) ~ p(x -1). But p(x-i) < px+p-x (if p> x) ad px+p-x=(p-i)(x+i)+i~s(px+l). 2 Therefore :3 p ~ x -1 so that S(pX) < S(px+l). Tha Fs(x) = S(p+ +S(pX)+ +S(p~X» Fs(x + 1) = S(ptl)+... +S(pX+l)+ +S(p~+;» > Fs(x) I coclusio Fs ( x + 1) > Fs (x) for x ad x + 1 composite umbers. If x is a prime umber 7Z{ x) = 7Z{ x + 1) ad the fact that the equatio Fs ( x) = Fs ( x + 1) has o solutio has the same demostratio as above. Fially the equatio Fs (x) = Fs (x + 1) has o solutio for ay x EN - {O, I}. PROPOSITION 3. The fuctio Fs(x) is strictly icreasig fuctio o its domai of defiitio. The proof of this property is justified by the propositio 2. PROPOSITION 4. Fs(x + y) > Fs(x) + Fs(Y) Vx,y EN - {O, I}. Proof Let X,Y EN - {O, I} ad we suppose x < y. Accordig to the defiitio of FS(x) we have: 7

3 F( ) S( x+y). S(,rTY)... S( X"' Y ). S(.r"'V) X + Y = PI T"'T P:r(x) ' P:r(.r)+!'... + P:ft~) T (4) 'S(.r+Y) S(.r+Y ) -r P1f(y) P1f(x+y) But from (1) we have the followig iequalities: A = (x + Y)(Pt P:r(.r) + P:r(x)+l P(x+y) - ;r(x + y» < F(x + y) $ ad X(PI P1f(x) - ;r(x» + Y(PI P:r(.r) +"'+P:r(X) +"'+P(y) - ;r(y» < F(x) + F(y) $ $ x(p,.+ + P1f(X» + yep, P,t(x) + P1l(x)+i ' ~.» = B (6) We proof that B < A. B < A <:::> x(p, P1l(r» + Y(A P1l(r» + Y(PJZ(r) PJZ(y») < x(a P:r(.r» + Y(A+"'~ P1f(.r» + x(p1l(r)+l P:r(r+y» - x(x + y) + + Y(P1l(x)+l P1l(y» + y( P.t(y)+l P1l(x+y» - y;r(x + y) <:::> x(p1l(r)+l+"'+p:r(x+y) - ;r(x+ y»+ Y(p,t(Y)+i+... P1E{r+y) - ;r(x+ y» > But P:r(r+y) ~ ;r(x + y) so that the iequality from above is true. CONSEQUENCE: FS(XY) > Fs(x) + Fs(Y) 'if x,y en -'{O,l} Because x ady en - {O,l} ad xy > x + Y tha Fs(XY) > Fs(x + y) > Fs(x) + Fs(Y) PROPOsmON 5. We try to fid lim FS() -+oo a We have Fs() = L S(pr') ad: O<p;S P!=p1fll! A + p,,+"'+p:() - ;r() a - I PI + P2+"'+P1r(II) a - 1 If a= 1 tha lim 1 - a (A+"'+p1l(II)-;r(» = lim (Pt+"'+p1f(II)-;r(» = +co ~ lim Fs(~) =+00 -+OO 11-+"" _ ~ 8

4 We cosider ow a > 1. We try to fid lim --"I=::.JI~ a - I ;'!() L P, - ;r() L P,!!( ) Let a = L P, - ;r() ad bl1 = a - 1 1=1 ad lim ~ applig Stolz - Cesaro: -+oc a - I :,(+l) L PI-;r(+l)- L PI+;r() 1=1 1=1 = ( + l)a-l _ a - I if ( + 1) is a prime 0, otherwise Let c = L Pi ad d = a - I. 1=1 Tha!!(+l) L PI-LPI C+1 - c = 1=1 1=1 = P!!(+I) = d + 1 -d ( + l)a-i _ a- I ( + l)a-i - a - I +1 if ( + l)a-1 _ a - I ( + I) is a prime 0, otherwise First we cosider the limit of the fuctio. lim x I 1 Co r-+co (x + It-I - x a - I = 1m = 0 lor a - 2 > 1 HCO ( a-l)[ (x + l)a-2 - x a - 2 ] We used the l'hospital theorem: I the same way we have lim x + 1 = 0 for a > 3. r-+co (x + l)a-i - x a - I So, for a > 3 we have:. PI+P2+"'+P-;r() ad hm a-i = 0 r-+co r PI + P P - 0 1m a-i -. r-+co So lim F() = o. r-+co a Fially lim F (:) = { 0 r-+"" +00 for a> 3 for a ~ 1 9

5 BmuOGRAPHY II) M. Adrei, C. Dumitrescu., V. Seleacu, L. T U\escu, ~ t Zafir [2} P. Groas Some remarks o the Smaradache Fuctio, Smaradache Fuctio Joural. Vol. 4, NO.1 (1994) 1-5; A ote o S(p), Smaradache Fuctio Joural. V. 2-3, No.1 (1993) 33~ (3) M. Adrei. 1. BaIaceoiu. C Dumitrescu., E.lUdescu. N. Radescu. V. Seleacu [4) F. Smaradache A liear combiatio with Smaradache Fuctio to obtai the Idetity, Proceedigs of 26 m Aual Iraia Mathematic Coferece Sbahid Babarar Uiversity of Kerma Kerma - Ira March A Fuctio i the Number Theory, A. Uiv. Tui~ Ser.StMat. VotXVIII, fase. 1(1980) 9,

Bertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n.

Bertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n. Bertrad s Postulate Our goal is to prove the followig Theorem Bertrad s Postulate: For every positive iteger, there is a prime p satisfyig < p We remark that Bertrad s Postulate is true by ispectio for,,